Write the eccentricity of the ellipse 9x^2+5y^2-18x-2y-16=0.

Question

Write the eccentricity of the ellipse $9\text{x}^2+5\text{y}^2-18\text{x}-2\text{y}-16=0.$

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Answer

$9\text{x}^2+5\text{y}^2-18\text{x}-2\text{y}-16=0.$
$\Rightarrow9\big(\text{x}^2-2\text{x}\big)+5\big(\text{y}^2-\frac{2\text{y}}{5}\big)=16$
$\Rightarrow9(\text{x}^2-2\text{x}+1)+5\big(\text{y}^2-\frac{2\text{y}}{5}+\frac{1}{25}\big)=16+9+\frac{1}{5}$
$9(\text{x}-1)^2+5\big(\text{y}-\frac{1}{5}\big)^2=\frac{126}{5}$
$\frac{(\text{x}-1)^2}{\frac{14}{5}}+\frac{\big(\text{y}-\frac{1}{5}\big)^2}{\frac{126}{25}}=1$
$\Rightarrow\text{a}^2=\frac{126}{45}$ and $\text{b}^2=\frac{126}{25}$
Clearly, a < b
Now, $\text{e}=\sqrt{1-\frac{\text{a}^2}{\text{b}^2}}$
$\Rightarrow\text{e}=\sqrt{1-\frac{\frac{126}{45}}{\frac{126}{25}}}$
$\Rightarrow\text{e}=\sqrt{1-\frac{5}{9}}$
$\Rightarrow\text{e}=\frac{2}{3}$

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